Subgroups — #31012617

Order 17152.

646 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22573352	C₂	1	2	yes	maximal, complemented
2	#22571997	C₂	1	2	yes	maximal, complemented
3	#22562470	C₂	1	1	yes	maximal, characteristic, complemented
4	#22562340	C₂	1	2	yes	maximal, complemented
5	#22562278	C₂	1	2	yes	maximal, complemented
6	#22562188	C₂	1	2	yes	maximal, complemented
7	#22562167	C₂	1	2	yes	maximal, complemented
8	#22562047	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2178	C₂	1	1	yes	maximal, characteristic, complemented
10	#16864210	C₂²	1	2	yes	—
11	#16864208	—	2	2	no	—
12	#16864198	C₂²	1	2	yes	complemented
13	#16863838	—	2	2	no	—
14	#16863836	C₂²	1	2	yes	complemented
15	#16863831	C₂²	1	2	yes	—
16	#16863830	—	2	2	no	—
17	#16863827	—	2	2	no	—
18	C₂×Dic₅₃₆	C₂²	1	1	yes	characteristic, complemented
19	C₂×#14523988	C₂²	1	2	yes	complemented
20	SD₃₂×D₆₇	—	2	2	no	—
21	C₂×D₁₀₇₂	C₂²	1	1	yes	characteristic
22	#16863812	C₂²	1	2	yes	complemented
23	#16863780	C₂²	1	2	yes	complemented
24	#16863672	C₂²	1	2	yes	complemented
25	#16863667	C₂²	1	2	yes	—
26	#16863659	—	2	2	no	—
27	#16863575	—	2	2	no	—
28	#16863570	—	2	2	no	—
29	#16863514	C₂²	1	1	yes	characteristic
30	#16863217	C₂²	1	1	yes	characteristic, complemented
31	#16863211	C₂²	1	2	yes	complemented
32	#16863208	C₂²	1	2	yes	complemented
33	#16863179	C₂²	1	2	yes	complemented
34	#508×C₆₇	C₂²	1	1	yes	characteristic
35	#443×C₆₇	C₂²	1	2	yes	complemented
36	#368×C₆₇	C₂²	1	1	yes	characteristic, complemented
37	#359×C₆₇	C₂²	1	2	yes	complemented
38	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic, complemented
39	#14524090	—	2	2	no	—
40	#14524089	—	2	2	no	—
41	#14524076	—	2	2	no	—
42	#14524076	—	4	2	no	—
43	#14524073	—	2	2	no	—
44	#14524072	—	4	2	no	—
45	#14524071	—	2	2	no	—
46	#14524071	—	4	2	no	—
47	#14524070	—	4	2	no	—
48	#14524068	—	2	2	no	—
49	#14524066	—	2	2	no	—
50	#14524065	—	4	2	no	—
51	#14524065	D₄	1	1	yes	characteristic
52	#14524062	—	2	2	no	—
53	C₂×Dic₂₆₈	C₂³	1	2	yes	—
54	Q₁₆×D₆₇	—	2	2	no	—
55	C₂×#11761617	—	2	2	no	—
56	C₂×#11761617	—	4	2	no	—
57	C₂×D₅₃₆	C₂³	1	2	yes	—
58	D₈×D₆₇	—	2	2	no	—
59	#14524049	C₂³	1	2	yes	—
60	#14524031	—	2	2	no	—
61	Dic₅₃₆	—	4	1	no	—
62	#14523989	—	2	2	no	—
63	#14523988	—	2	2	no	—
64	#14523988	—	2	2	no	—
65	#14523987	—	2	2	no	—
66	#14523986	—	2	2	no	—
67	D₁₀₇₂	—	4	1	no	—
68	#14523984	—	2	2	no	—
69	#14523983	C₂³	1	2	yes	—
70	#14523982	—	2	2	no	—
71	C₁₆×D₆₇	—	2	2	no	—
72	#14523979	D₄	1	1	yes	characteristic
73	#14523971	D₄	1	2	yes	—
74	#14523961	—	2	2	no	—
75	#14523959	—	2	2	no	—
76	#135×C₆₇	C₂³	1	2	yes	—
77	#131×C₆₇	—	2	2	no	—
78	Q₃₂×C₆₇	D₄	1	1	yes	characteristic
79	SD₃₂×C₆₇	D₄	1	2	yes	—
80	D₁₆×C₆₇	D₄	1	1	yes	characteristic
81	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
82	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
83	#108×C₆₇	C₂³	1	2	yes	—
84	#104×C₆₇	—	2	2	no	—
85	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
86	#11761629	—	4	2	no	—
87	#11761628	—	4	2	no	—
88	#11761628	—	4	2	no	—
89	#11761628	C₂×D₄	1	2	yes	—
90	#11761627	—	4	2	no	—
91	#11761627	—	8	2	no	—
92	C₂×Dic₁₃₄	—	2	2	no	—
93	C₂×Dic₁₃₄	—	4	2	no	—
94	Q₈×D₆₇	—	4	2	no	—
95	C₂×D₂₆₈	—	2	2	no	—
96	C₂×D₂₆₈	—	4	2	no	—
97	D₄×D₆₇	—	4	2	no	—
98	D₄×D₆₇	—	8	2	no	—
99	Dic₂₆₈	—	2	2	no	—
100	Dic₂₆₈	—	2	2	no	—
101	Dic₂₆₈	—	4	2	no	—
102	Dic₂₆₈	D₈	1	1	yes	characteristic
103	#11761618	—	4	2	no	—
104	#11761617	—	4	2	no	—
105	#11761617	—	4	2	no	—
106	#11761617	—	4	2	no	—
107	#11761617	—	4	2	no	—
108	#11761617	D₈	1	2	yes	—
109	#11761616	—	4	2	no	—
110	#11761615	—	4	2	no	—
111	D₅₃₆	—	2	2	no	—
112	D₅₃₆	—	2	2	no	—
113	D₅₃₆	—	4	2	no	—
114	D₅₃₆	D₈	1	1	yes	characteristic
115	#11761613	—	4	2	no	—
116	#11761612	C₂×D₄	1	2	yes	—
117	#11761611	—	2	2	no	—
118	C₈×D₆₇	—	2	2	no	—
119	(C₂×D₆₇)⋊C₄	—	8	2	no	—
120	C₆₇⋊C₁₆	—	2	2	no	—
121	#41×C₆₇	—	2	2	no	—
122	Q₁₆×C₆₇	C₂×D₄	1	2	yes	—
123	SD₁₆×C₆₇	—	2	2	no	—
124	D₈×C₆₇	C₂×D₄	1	2	yes	—
125	M₁₆×C₆₇	—	2	2	no	—
126	M₁₆×C₆₇	—	4	2	no	—
127	C₂×C₈×C₆₇	—	4	2	no	—
128	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
129	C₂×C₈×C₆₇	C₂⁴	1	1	yes	characteristic
130	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
131	C₁₆×C₆₇	—	2	1	no	—
132	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
133	C₁₆×C₆₇	C₂×D₄	1	1	yes	characteristic
134	C₂²×D₆₇	—	4	2	no	—
135	C₂²×D₆₇	—	8	2	no	—
136	Q₈×C₆₇	—	2	2	no	—
137	D₄×C₆₇	—	2	2	no	—
138	D₄×C₆₇	—	4	2	no	—
139	C₂×C₄×C₆₇	—	2	1	no	—
140	C₂×C₄×C₆₇	—	4	2	no	—
141	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
142	C₂²⋊D₆₇	—	2	2	no	—
143	C₂²⋊D₆₇	—	8	2	no	—
144	C₂²⋊D₆₇	—	8	2	no	—
145	C₂×Dic₆₇	—	4	2	no	—
146	C₂×Dic₆₇	—	8	2	no	—
147	D₂₆₈	—	4	2	no	—
148	D₂₆₈	—	4	2	no	—
149	D₂₆₈	—	8	2	no	—
150	D₂₆₈	C₂×D₈	1	2	yes	—
151	C₄×D₆₇	—	2	2	no	—
152	C₄×D₆₇	—	8	2	no	—
153	C₄×D₆₇	—	8	2	no	—
154	Dic₁₃₄	—	4	2	no	—
155	Dic₁₃₄	—	4	2	no	—
156	Dic₁₃₄	—	8	2	no	—
157	Dic₁₃₄	C₂×D₈	1	2	yes	—
158	C₈×C₆₇	—	4	2	no	—
159	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
160	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
161	C₈×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
163	C₆₇⋊C₈	—	2	2	no	—
164	C₂²×C₆₇	—	4	2	no	—
165	C₂²×C₆₇	#576	1	1	yes	characteristic
166	C₂×D₆₇	—	2	2	no	—
167	C₂×D₆₇	—	8	2	no	—
168	C₂×D₆₇	—	16	2	no	—
169	C₂×D₆₇	—	32	2	no	—
170	C₄×C₆₇	—	4	1	no	—
171	C₄×C₆₇	—	4	2	no	—
172	C₄×C₆₇	D₄²	1	1	yes	characteristic
173	C₄×C₆₇	C₂²×D₈	1	1	yes	characteristic
174	Dic₆₇	—	2	2	no	—
175	Dic₆₇	—	8	2	no	—
176	#33857	—	67	1	no	maximal, Sylow
177	C₂×C₆₇	—	2	1	no	—
178	C₂×C₆₇	—	8	2	no	—
179	C₂×C₆₇	D₄×D₈	1	1	yes	characteristic
180	D₆₇	—	4	2	no	—
181	D₆₇	—	16	2	no	—
182	#3416	—	67	2	no	—
183	#3290	—	67	2	no	—
184	#2249	—	67	1	no	—
185	#2223	—	67	2	no	—
186	#2212	—	67	2	no	—
187	#2195	—	67	2	no	—
188	#2192	—	67	2	no	—
189	#2178	D₆₇	1	1	yes	characteristic, complemented
190	#2170	—	67	1	no	—
191	C₆₇	#33857	1	1	yes	characteristic, complemented, Sylow
192	#578	—	67	2	no	—
193	#577	—	134	2	no	—
194	#576	—	67	2	no	—
195	#510	—	67	2	no	—
196	#510	—	134	2	no	—
197	#509	—	67	2	no	—
198	#509	—	134	2	no	—
199	#508	—	134	2	no	—
200	#508	C₂×D₆₇	1	1	yes	characteristic
201	C₂×Q₃₂	—	67	1	no	—
202	C₂×SD₃₂	—	67	2	no	—
203	C₂×SD₃₂	—	134	2	no	—
204	C₂×D₁₆	—	67	1	no	—
205	#504	—	67	2	no	—
206	#495	—	67	2	no	—
207	#473	—	67	2	no	—
208	#472	—	67	2	no	—
209	#471	—	134	2	no	—
210	#455	—	134	2	no	—
211	#454	—	134	2	no	—
212	#443	—	67	1	no	—
213	#443	C₂×D₆₇	1	2	yes	complemented
214	#368	C₂×D₆₇	1	1	yes	characteristic, complemented
215	#364	—	67	1	no	—
216	#362	—	67	2	no	—
217	#361	—	67	2	no	—
218	#359	C₂×D₆₇	1	2	yes	complemented
219	#350	—	67	2	no	—
220	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic, complemented
221	2₋¹⁺⁴	—	134	2	no	—
222	H(5,𝔽₂)	—	134	2	no	—
223	#137	—	268	2	no	—
224	#137	—	268	2	no	—
225	#137	—	134	2	no	—
226	#137	—	134	2	no	—
227	AGL(1,ℤ/8)	—	268	2	no	—
228	AGL(1,ℤ/8)	—	268	2	no	—
229	AGL(1,ℤ/8)	—	134	2	no	—
230	AGL(1,ℤ/8)	—	134	2	no	—
231	#135	—	268	2	no	—
232	#135	—	67	1	no	—
233	#135	—	134	2	no	—
234	#135	—	134	2	no	—
235	#135	C₂²×D₆₇	1	2	yes	—
236	C₂×Q₁₆	—	67	2	no	—
237	C₂×Q₁₆	—	134	2	no	—
238	C₂×SD₁₆	—	268	2	no	—
239	C₂×SD₁₆	—	134	2	no	—
240	C₂×D₈	—	67	2	no	—
241	C₂×D₈	—	134	2	no	—
242	#131	—	2	2	no	—
243	#131	—	67	2	no	—
244	#124	—	134	2	no	—
245	Q₃₂	—	268	1	no	—
246	Q₃₂	—	134	2	no	—
247	Q₃₂	D₂₆₈	1	1	yes	characteristic
248	SD₃₂	—	134	2	no	—
249	SD₃₂	—	134	2	no	—
250	SD₃₂	—	134	2	no	—
251	SD₃₂	—	134	2	no	—
252	SD₃₂	D₂₆₈	1	2	yes	—
253	D₁₆	—	268	1	no	—
254	D₁₆	—	134	2	no	—
255	D₁₆	D₂₆₈	1	1	yes	characteristic
256	M₃₂	—	67	2	no	—
257	M₃₂	—	134	2	no	—
258	C₂×C₁₆	—	134	2	no	—
259	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
260	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
261	#108	—	67	1	no	—
262	#108	C₂²×D₆₇	1	2	yes	—
263	#104	—	2	2	no	—
264	#104	—	67	2	no	—
265	#101	—	134	2	no	—
266	#100	—	134	2	no	—
267	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
268	#41	—	2	2	no	—
269	#41	—	268	2	no	—
270	#41	—	268	2	no	—
271	#41	—	268	2	no	—
272	#41	—	268	2	no	—
273	#41	—	536	2	no	—
274	#41	—	67	2	no	—
275	C₂×Q₈	—	268	2	no	—
276	C₂×Q₈	—	268	2	no	—
277	C₂×Q₈	—	134	2	no	—
278	C₂×D₄	—	268	2	no	—
279	C₂×D₄	—	268	2	no	—
280	C₂×D₄	—	536	2	no	—
281	C₂×D₄	—	134	2	no	—
282	Q₁₆	—	268	2	no	—
283	Q₁₆	—	268	2	no	—
284	Q₁₆	—	67	1	no	—
285	Q₁₆	—	134	2	no	—
286	Q₁₆	—	134	2	no	—
287	Q₁₆	C₂×D₂₆₈	1	2	yes	—
288	SD₁₆	—	2	2	no	—
289	SD₁₆	—	268	2	no	—
290	SD₁₆	—	268	2	no	—
291	SD₁₆	—	268	2	no	—
292	SD₁₆	—	268	2	no	—
293	SD₁₆	—	268	2	no	—
294	SD₁₆	—	268	2	no	—
295	SD₁₆	—	67	2	no	—
296	D₈	—	268	2	no	—
297	D₈	—	268	2	no	—
298	D₈	—	67	1	no	—
299	D₈	—	134	2	no	—
300	D₈	—	134	2	no	—
301	D₈	C₂×D₂₆₈	1	2	yes	—
302	M₁₆	—	2	2	no	—
303	M₁₆	—	4	2	no	—
304	M₁₆	—	67	2	no	—
305	M₁₆	—	134	2	no	—
306	C₂×C₈	—	4	2	no	—
307	C₂×C₈	—	134	2	no	—
308	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
309	C₂×C₈	C₂³×D₆₇	1	1	yes	characteristic
310	C₂²⋊C₄	—	536	2	no	—
311	C₄²	D₄×D₆₇	1	1	yes	characteristic
312	C₁₆	—	2	1	no	—
313	C₁₆	—	134	2	no	—
314	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
315	C₁₆	C₂×D₂₆₈	1	1	yes	characteristic
316	C₂³	—	268	2	no	—
317	C₂³	—	536	2	no	—
318	Q₈	—	2	2	no	—
319	Q₈	—	268	2	no	—
320	Q₈	—	268	2	no	—
321	Q₈	—	536	2	no	—
322	Q₈	—	67	2	no	—
323	D₄	—	2	2	no	—
324	D₄	—	4	2	no	—
325	D₄	—	268	2	no	—
326	D₄	—	268	2	no	—
327	D₄	—	536	2	no	—
328	D₄	—	536	2	no	—
329	D₄	—	536	2	no	—
330	D₄	—	67	2	no	—
331	D₄	—	134	2	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	4	2	no	—
334	C₂×C₄	—	268	2	no	—
335	C₂×C₄	—	536	2	no	—
336	C₂×C₄	—	536	2	no	—
337	C₂×C₄	—	536	2	no	—
338	C₂×C₄	—	134	2	no	—
339	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
340	C₈	—	4	2	no	—
341	C₈	—	134	2	no	—
342	C₈	D₈×D₆₇	1	1	yes	characteristic
343	C₈	D₈×D₆₇	1	1	yes	characteristic
344	C₈	C₂²×D₂₆₈	1	1	yes	characteristic
345	C₈	#14524089	1	1	yes	characteristic
346	C₂²	—	4	2	no	—
347	C₂²	—	536	2	no	—
348	C₂²	—	1072	2	no	—
349	C₂²	—	2144	2	no	—
350	C₂²	—	134	2	no	—
351	C₂²	#16864198	1	1	yes	characteristic
352	C₄	—	4	1	no	—
353	C₄	—	4	2	no	—
354	C₄	—	536	2	no	—
355	C₄	—	134	2	no	—
356	C₄	D₄×D₂₆₈	1	1	yes	characteristic
357	C₄	C₂×D₈×D₆₇	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	8	2	no	—
360	C₂	—	268	2	no	—
361	C₂	—	1072	2	no	—
362	C₂	D₈×D₂₆₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂	D₈×D₂₆₈
commutator, cosocle	C₂×C₈×C₆₇	C₂⁴
Frattini	C₂×C₈	C₂³×D₆₇
Fitting	C₆₇×#2178	C₂
socle	C₂×C₆₇	D₄×D₈
hypercenter	#2178	D₆₇
hypocenter	C₆₇	#33857
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #2178

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #22573352	C₂	semidirect
2. #22571997	C₂	semidirect
3. #22562470	C₂	semidirect
4. #22562340	C₂	semidirect
5. #22562278	C₂	semidirect
6. #22562188	C₂	semidirect
7. #22562167	C₂	semidirect
8. #22562047	C₂	semidirect
9. C₆₇×#2178	C₂	semidirect
12. #16864198	C₂²	semidirect
14. #16863836	C₂²	semidirect
18. C₂×Dic₅₃₆	C₂²	semidirect
19. C₂×#14523988	C₂²	semidirect
22. #16863812	C₂²	semidirect
23. #16863780	C₂²	semidirect
24. #16863672	C₂²	semidirect
30. #16863217	C₂²	semidirect
31. #16863211	C₂²	semidirect
32. #16863208	C₂²	semidirect
33. #16863179	C₂²	semidirect
35. #443×C₆₇	C₂²	semidirect
36. #368×C₆₇	C₂²	semidirect
37. #359×C₆₇	C₂²	semidirect
38. C₄×C₁₆×C₆₇	C₂²	semidirect
189. #2178	D₆₇	semidirect
191. C₆₇	#33857	semidirect
213. #443	C₂×D₆₇	semidirect
214. #368	C₂×D₆₇	semidirect
218. #359	C₂×D₆₇	semidirect
220. C₄×C₁₆	C₂×D₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22573352	2
#22571997	2
#22562470	2
#22562340	2
#22562278	2
#22562188	2
#22562167	2
#22562047	2
C₆₇×#2178	2
#33857	67